The Greek alphabet is adapted from the consonantal writing systems of the Levant, and I used to have a vague idea that Greek got its vowel signs by adapting signs for Semitic consonants not present in Greek. Greek has no aspirated “s”, for example, so Greeks using the Northwest Semitic abjad to write Greek found that they did not need the sign shin ש for transcribing Greek consonants and could use it for something else. As I learn a bit of Aramaic I realize that the process was much more straightforward.
The Northwest Semitic signs aleph א, he ה, and yod י already served double duty as both the consonants ˀ h and y and the vowels a, e, and i. These so-called matres lectionis usually had a vocalic reading at the end of a word and a consonantal reading elsewhere. It was therefore very natural to turn them into the Greek letters alpha Α, epsilon Ε, and iota Ι representing English a, e, and i. The Attic form of the Greek alphabet created a new sign omega Ω for the vowel in English “saw” and turned Aramaic chet ח (a strong “h” sound missing in Attic) into eta H (the vowel in English “say”) and Aramaic ayin into omicron Ο. The Attic script also fixed the value of the vowel signs (most matres lectionis can represent two or three different vowels depending on context and the scribe’s preferences). Yet every child who learned to write one of the Northwest Semitic abjads learned that signs could represent vowels as well as consonants, and three of the six Attic vowels could also be used for the same vowel in the Semitic languages.
The Northwest Semitic signs waw ו and yod י were also used as matres lectionis in addition to their consonantal values w and y. Linguists call these sounds semivowels because they share some properties of the consonants and of the vowels (remember the nursery rhyme “A E I O U and sometimes Y and W”?). Although Greek iota I is always a vowel and digamma F is always a consonant, Roman scribes returned to the old Northwest Semitic practice. Latin I could represent both English I and J (really a “y” sound) and Latin V English W and U. A consistent use of I and J and of W V U to represent different sounds appears quite late in the history of the Latin alphabet.
Further Reading: Peter T. Daniels, “Scripts of Semitic Languages,” in Robert Hetzron ed., The Semitic Languages (Routledge, 1997) p. 22, S. Segert, Altaramäische Grammatik (Lepizig 1975), Ancient Scripts “Aramaic”, Arshama Project, Aramaic Skeleton Grammar 01 (Overview, Alphabet).
Edit 2015-03-28: Corrected a typo and changed West Semitic to Northwest Semitic for any fussy Semiticists who drop by (historical linguists call all the Semitic languages from Syria to Ethiopia West Semitic, and group languages like Hebrew Aramaic and Ugaritic under the label Northwest Semitic; East Semitic is mainly Akkadian)